Topological Structure of Quantum Algorithms

Abstract

We use a topological formalism to examine the Deutsch-Jozsa, hidden subgroup and Grover algorithms. This reveals important structures hidden by conventional algebraic presentations, and allows short and visual proofs of correctness via local topological operations. The resulting transparency makes generalizations of these algorithms clear, which gives rise to simpler descriptions of generalized Deutsch-Jozsa and hidden subgroup algorithms already in the literature, and a new generalization of Grover's algorithm.